# Erdos conjecture, number of cliques, Turan`s graph

Erdos&Stone conjectured in 1946-08 that

there are at least $ck-1$ (k+1)-cliques in $G=(V,E)$ whenever $|V|=ck$, $|E|-1$ is the edge-count in Turan's $T(|V|,k)$, i.e.,

$|E|= 1+\lfloor {(k-1)|V|^2} / (2k) \rfloor$.

I am looking to see whether this conjecture has been proved in literature. Didn't see a trace of it myself.

Isn't the conjecture in the paper that the number of $(k+1)$-cliques in this case is at least $c^{k-1}$? That's the number you get by adding a single edge to the Turan-graph. I think it's proved in 1969-10. Also relevant: 1983-28. Recently, there has been significant progress on bounding the number of cliques as a function of the quadratic term of the number of edges: arxiv:1212.2454.